Problem: Simplify and expand the following expression: $ \dfrac{2q + 1}{2q - 6}+\dfrac{-2}{q - 3} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2q - 6)(q - 3)$ Multiply the first term by $\dfrac{q - 3}{q - 3}$ $ \begin{align*} \dfrac{2q + 1}{2q - 6} \times \dfrac{q - 3}{q - 3} & = \dfrac{(2q + 1)(q - 3)}{(2q - 6)(q - 3)} \\ & = \dfrac{2q^2 - 5q - 3}{(2q - 6)(q - 3)}\end{align*} $ Multiply the second term by $\dfrac{2q - 6}{2q - 6}$ $ \begin{align*} \dfrac{-2}{q - 3} \times \dfrac{2q - 6}{2q - 6} & = \dfrac{(-2)(2q - 6)}{(q - 3)(2q - 6)} \\ & = \dfrac{-4q + 12}{(q - 3)(2q - 6)}\end{align*} $ Now we have: $ = \dfrac{2q^2 - 5q - 3}{(2q - 6)(q - 3)} + \dfrac{-4q + 12}{(q - 3)(2q - 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2q^2 - 5q - 3 - 4q + 12}{(2q - 6)(q - 3)} $ $ = \dfrac{2q^2 - 9q + 9}{(2q - 6)(q - 3)}$ Expand the denominator: $ = \dfrac{2q^2 - 9q + 9}{2q^2 - 12q + 18}$